Figure REF  shows the averaged game graphWe use the software Gephi [1]} for all graph visualizations. for \(p=0.75\) , where the nodes' sizes and colors represent the \(\alpha \) -Rank-collection. One can see that (Hawk, Hawk) dominates the distribution. However, the edges pointing towards (Dove, Hawk) and (Hawk, Dove) show small expected improvements, so these strategies still appear in the long run.
Note that for \(p \ge \frac{1}{2}\)  the Bayesian Nash equilibrium of this game would be the pure strategy with playing Hawk as PD type and Dove as AC type. On the other hand, an equilibrium with distributional strategies exists only for the case of \(p < \frac{1}{2}\) , together with two pure strategy equilibria. This again highlights the NE's difference to \(\alpha \) -Rank-collections. Instead of having best-responses to strategies conditioned on the individual types that constitute a BNE, an \(\alpha \) -Rank-collection captures variances in strategic play resulting from uncertainties in the players' vNM functions.
